Elements of Quotients of Univariate Polynomial Rings#
EXAMPLES: We create a quotient of a univariate polynomial ring over \(\ZZ\).
sage: R.<x> = ZZ[]
sage: S.<a> = R.quotient(x^3 + 3*x -1)
sage: 2 * a^3
-6*a + 2
Next we make a univariate polynomial ring over \(\ZZ[x]/(x^3+3x-1)\).
sage: S1.<y> = S[]
And, we quotient out that by \(y^2 + a\).
sage: T.<z> = S1.quotient(y^2+a)
In the quotient \(z^2\) is \(-a\).
sage: z^2
-a
And since \(a^3 = -3x + 1\), we have:
sage: z^6
3*a - 1
sage: R.<x> = PolynomialRing(Integers(9))
sage: S.<a> = R.quotient(x^4 + 2*x^3 + x + 2)
sage: a^100
7*a^3 + 8*a + 7
sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = R.quotient(x^3-2)
sage: a
a
sage: a^3
2
For the purposes of comparison in Sage the quotient element \(a^3\) is equal to \(x^3\). This is because when the comparison is performed, the right element is coerced into the parent of the left element, and \(x^3\) coerces to \(a^3\).
sage: a == x
True
sage: a^3 == x^3
True
sage: x^3
x^3
sage: S(x^3)
2
AUTHORS:
William Stein
- class sage.rings.polynomial.polynomial_quotient_ring_element.PolynomialQuotientRingElement(parent, polynomial, check=True)#
Bases:
sage.rings.polynomial.polynomial_singular_interface.Polynomial_singular_repr
,sage.structure.element.CommutativeRingElement
Element of a quotient of a polynomial ring.
EXAMPLES:
sage: P.<x> = QQ[] sage: Q.<xi> = P.quo([(x^2+1)]) sage: xi^2 -1 sage: singular(xi) xi sage: (singular(xi)*singular(xi)).NF('std(0)') -1
- charpoly(var)#
The characteristic polynomial of this element, which is by definition the characteristic polynomial of right multiplication by this element.
INPUT:
var
- string - the variable name
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quo(x^3 -389*x^2 + 2*x - 5) sage: a.charpoly('X') X^3 - 389*X^2 + 2*X - 5
- fcp(var='x')#
Return the factorization of the characteristic polynomial of this element.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5) sage: a.fcp('x') x^3 - 389*x^2 + 2*x - 5 sage: S(1).fcp('y') (y - 1)^3
- field_extension(names)#
Given a polynomial with base ring a quotient ring, return a 3-tuple: a number field defined by the same polynomial, a homomorphism from its parent to the number field sending the generators to one another, and the inverse isomorphism.
INPUT:
names
- name of generator of output field
OUTPUT:
field
homomorphism from self to field
homomorphism from field to self
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<alpha> = R.quotient(x^3-2) sage: F.<a>, f, g = alpha.field_extension() sage: F Number Field in a with defining polynomial x^3 - 2 sage: a = F.gen() sage: f(alpha) a sage: g(a) alpha
Over a finite field, the corresponding field extension is not a number field:
sage: R.<x> = GF(25,'b')['x'] sage: S.<a> = R.quo(x^3 + 2*x + 1) sage: F.<b>, g, h = a.field_extension() sage: h(b^2 + 3) a^2 + 3 sage: g(x^2 + 2) b^2 + 2
We do an example involving a relative number field:
sage: R.<x> = QQ['x'] sage: K.<a> = NumberField(x^3-2) sage: S.<X> = K['X'] sage: Q.<b> = S.quo(X^3 + 2*X + 1) sage: F, g, h = b.field_extension('c')
Another more awkward example:
sage: R.<x> = QQ['x'] sage: K.<a> = NumberField(x^3-2) sage: S.<X> = K['X'] sage: f = (X+a)^3 + 2*(X+a) + 1 sage: f X^3 + 3*a*X^2 + (3*a^2 + 2)*X + 2*a + 3 sage: Q.<z> = S.quo(f) sage: F.<w>, g, h = z.field_extension() sage: c = g(z) sage: f(c) 0 sage: h(g(z)) z sage: g(h(w)) w
AUTHORS:
Craig Citro (2006-08-06)
William Stein (2006-08-06)
- is_unit()#
Return
True
ifself
is invertible.Warning
Only implemented when the base ring is a field.
EXAMPLES:
sage: R.<x> = QQ[] sage: S.<y> = R.quotient(x^2 + 2*x + 1) sage: (2*y).is_unit() True sage: (y+1).is_unit() False
- lift()#
Return lift of this polynomial quotient ring element to the unique equivalent polynomial of degree less than the modulus.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3-2) sage: b = a^2 - 3 sage: b a^2 - 3 sage: b.lift() x^2 - 3
- list(copy=True)#
Return list of the elements of
self
, of length the same as the degree of the quotient polynomial ring.EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3 + 2*x - 5) sage: a^10 -134*a^2 - 35*a + 300 sage: (a^10).list() [300, -35, -134]
- matrix()#
The matrix of right multiplication by this element on the power basis for the quotient ring.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3 + 2*x - 5) sage: a.matrix() [ 0 1 0] [ 0 0 1] [ 5 -2 0]
- minpoly()#
The minimal polynomial of this element, which is by definition the minimal polynomial of right multiplication by this element.
- norm()#
The norm of this element, which is the determinant of the matrix of right multiplication by this element.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5) sage: a.norm() 5
- trace()#
The trace of this element, which is the trace of the matrix of right multiplication by this element.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ) sage: S.<a> = R.quotient(x^3 -389*x^2 + 2*x - 5) sage: a.trace() 389